Question

Consider the differential equation dy/dx = cos(xpi)(y-1)^2
a) There is a horizontal line with equation y = c that satisfies this differential equation. Find the value of c.

Answer 1

\[dy/dx= cos(\pi x)(y-1)^2\]

?

Answer 2

yes

Answer 3

\[\frac{dy}{(y-1)^2}=cos(\pi x)dx\]

integrate both sides

\[\frac{1}{1-y}= \frac{1}{\pi}sin(\pi x) +C\]

Answer 4

I don't understand the y = c part satisfies the equation. What does that mean?

Answer 5

Can you solve the equation imran wrote for y?

Answer 6

I just realized that this might be easier than what I was doing

y=c

dy/dx=0

0 = cos(x pi)(c-1)^2

Answer 7

and solving for c , we get

c=1 or c=-1

Answer 8

\[y=1-\frac{\pi}{\pi{c}+\sin(\pi{x})}.\]

Answer 9

imran's approach could be easier.

Answer 10

\[\text{Cos}[\pi x]-2 c \text{Cos}[\pi x]+c^2 \text{Cos}[\pi x]\text{==}0\]

\[1-2 c +c^2 \text{==}0\]

(c+1)(c-1)=0

c=1,-1

Answer 11

Never mind! I was stupid :D Go with imran.

Answer 12

Ah, I don't really get why we differentiated y = c. I understand plugging it in though.

Answer 13

because we wanted y'

Answer 14

So when they say that y = c satisfies the equation. It means that y' = the differential equation?

Answer 15

if y=constant

y' must be zero